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We study a critical Grushin-type equation by applying Lyapunov-Schmidt reduction method and obtain the existence of infinitely many positive multi-bubbling solutions with cylindrical symmetry. Particularly, since the concentration points here can be saddle points of some function related to the potential, we locate the concentration points of these solutions by local Pohozaev identities rather than estimating the derivatives of the reduced functional as usual. [ABSTRACT FROM AUTHOR]

We consider three kinds of anisotropic double phase problems with Dirichlet boundary condition. In two of them we have strong singularity and an unbounded coefficient. Using variational method together with truncation, comparison and approximation techniques, we prove existence and multiplicity theorems for our problem. [ABSTRACT FROM AUTHOR]

We consider the prescribed scalar curvature problem on S N Δ S N v - N (N - 2) 2 v + K ~ (y) v N + 2 N - 2 = 0 on S N , v > 0 in S N , under the assumptions that the scalar curvature K ~ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of normalized positive solutions to the following Schrödinger equation with a perturbation - Δ u + λ u = | u | p - 2 u + h (x) in R N , ∫ R N u 2 d x = a , u ∈ H 1 (R N). We treat two cases. First, for the mass-subcritical case 2 < p < 2 + 4 N , we show that there exists a global minimizer with negative energy for arbitrarily positive perturbation. Secondly, for the mass-supercritical case 2 + 4 N < p < 2 ∗ where 2 ∗ = 2 N N - 2 if N ⩾ 3 and 2 ∗ = + ∞ if N = 1 , 2 , when h is small radial positive function, we prove the existence of a mountain pass solution with positive energy. It seems this is the first contribution to the normalized solution for such a perturbed equation. [ABSTRACT FROM AUTHOR]

The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K ∞ (R + d) that was defined in (Bachar et al. 2002:41, 2002) for d = 2 and in (Bachar and Mâagli 9(2):153–192, 2005) for d ≥ 3 and adapted to study some nonlinear elliptic problems in the half space. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system: Δ u = λ f (x , u , v) , Δ v = μ g (x , u , v) in R + d , lim x → (ξ , 0) u (x) = a 1 ϕ 1 (ξ) , lim x → (ξ , 0) v (x) = a 2 ϕ 2 (ξ) for all ξ ∈ R d − 1 , lim x d → ∞ u (x) x d = b 1 , lim x d → ∞ v (x) x d = b 2 , where ϕ 1 and ϕ 2 are nontrivial nonnegative continuous functions on ∂ R + d = R d − 1 × { 0 } , a 1 , a 2 , b 1 , b 2 are nonnegative constants such that (a 1 + b 1) (a 2 + b 2) > 0 . The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f (x , u , v) = p (x) u α g 1 (v) and g (x , u , v) = q (x) g 2 (u) v β with α ≥ 1 , β ≥ 1 , g 1 , g 2 are continuous on [ 0 , ∞) , and p , q are nonnegative functions in K ∞ (R + d) . [ABSTRACT FROM AUTHOR]

We consider the determination of an unknown spatial-dependent coefficient for the pseudoparabolic system with Dirichlet boundary condition from nonlocal inversion input data. Based on the positive property of the solution to the direct problem, the uniqueness and the Lipschitz conditional stability of this nonlinear inverse problem are addressed in Hölder space, by the principle of contraction mapping for specially chosen weight function describing the nonlocal inversion input. Then, an iterative algorithm is established in terms of the fixed point equation for the unknown potential to construct the approximate solution to this inverse problem, with rigorous convergence analysis and the error estimate on the iterative sequence, which also leads to the optimal approximation error for choosing the stopping rule of the iteration process. The proposed iterative algorithm is compared with classical optimization scheme with regularizing term, showing the advantages of our scheme. Some numerical results are presented to validate our proposed scheme. [ABSTRACT FROM AUTHOR]

We study the following nonlinear mass supercritical Schrödinger–Poisson–Slater equation - ▵ u + μ u - γ (| x | - 1 ∗ | u | 2) u - a f (u) = 0 in R 3 satisfying the normalized constrain ∫ R 3 | u | 2 d x = m , and m > 0 prescribed. The nonlinearity f is super-critical. Under some mild assumptions, we establish the existence of ground state and infinitely many radial solutions by constructing a particular bounded Palais-Smale sequence when γ < 0 , a > 0 . Meanwhile, we obtain the non-existence result in the case γ < 0 , a < 0 and the existence result when γ > 0 , a < 0 via variational methods. [ABSTRACT FROM AUTHOR]

We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian (−Δ)s (0 < s < 1). First we obtain a Liouville type theorem in the often-used defining space . An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases. Considering the anti-symmetric property, one can naturally extend the defining space from to . Surprisingly, with this extension, we show the existence of non-trivial solutions. This is very different from the previous results of the Lane-Emden system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional pp-equation with a gradient term: (−Δ)psu(x)=f(x,u(x),∇u(x)).{\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional pp-laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933].

This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: (−Δ)su+V(x)u=∫RNA(εy)∣u(y)∣p∣x−y∣μdyA(εx)∣u(x)∣p−2u(x),x∈RN,{\left(-\Delta )}^{s}u+V\left(x)u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{A\left(\varepsilon y)| u(y){| }^{p}}{| x-y{| }^{\mu }}{\rm{d}}y\right)A\left(\varepsilon x)| u\left(x){| }^{p-2}u\left(x),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where s∈(0,1)s\in \left(0,1), N>2sN\gt 2s, 0

In this article, we consider the following Schrödinger-Poisson system: −Δu+u+k(x)ϕ(x)u=f(x)∣u∣p−1u+g(x),x∈R3,−Δϕ=k(x)u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+u+k\left(x)\phi \left(x)u=f\left(x)| u{| }^{p-1}u+g\left(x),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =k\left(x){u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. with p∈(3,5)p\in \left(3,5). Under suitable assumptions on potentials f(x)f\left(x), g(x)g\left(x) and k(x)k\left(x), then at least four positive solutions for the above system can be obtained for sufficiently small ‖g‖H−1(R3)\Vert g{\Vert }_{{H}^{-1}\left({{\mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.

In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P)−Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞,- \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0}\mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation - Δ p u = λ | u | p - 2 u + a (x) | u | q - 2 u in a bounded domain Ω ⊂ R N , where 1 < q < p , λ ∈ R , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in { x ∈ Ω : a (x) > 0 } , when the parameter λ lies in a neighborhood of the critical value λ ∗ : = inf ∫ Ω | ∇ u | p d x / ∫ Ω | u | p d x : u ∈ W 0 1 , p (Ω) \ { 0 } , ∫ Ω a | u | q d x ≥ 0 . Among main results, we show that if p > 2 q and either ∫ Ω a φ p q d x = 0 or ∫ Ω a φ p q d x > 0 is sufficiently small, then such solutions do exist in a right neighborhood of λ ∗ . Here φ p is the first eigenfunction of the Dirichlet p-Laplacian in Ω . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case q > p or in the sublinear case q < p = 2 the nonexistence takes place for any λ ≥ λ ∗ . Moreover, we prove that if p > 2 q and ∫ Ω a φ p q d x > 0 is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of λ ∗ , two of which are strictly positive in { x ∈ Ω : a (x) > 0 } . [ABSTRACT FROM AUTHOR]

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on R d. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to some power p > d. We establish new estimates for the Lp-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following Kirchhoff-type equation { − (a + b ∫ R 3 | ∇ u | 2 d x) △ u + V (x) u = f (u) , x ∈ R 3 ; u ∈ H 1 (R 3) , where a>0 and b>0 are constants. Suppose that the nonnegative continuous potential V is not an asymptotic constant at infinity, and f satisfies some relatively weak conditions in the absence of the usual Ambrosetti–Rabinowitz type condition or monotonicity condition on f (t) t 3 . The result of this paper can be applied to the case where f (t) = | t | p − 2 t with 2 < p ≤ 4. By using some new techniques and subtle analysis, we prove that the above problem admits at least one ground state solution. It is worth mention that our result generalize those obtained in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R 3 . J Differ Equ. 2014;257:566–600], Tang and Chen [Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differ Equ. 2017;56:110–134], Guo [Ground states for Kirchhoff equations without compact condition. J Differ Equ. 2015;259:2884–2902] and some other related literatures. In particular, we give a proof for the Pohožaev type identity associated with the above equation, when V is unbounded. [ABSTRACT FROM AUTHOR]

We construct a radial smooth positive ancient solution for energy critical semi-linear heat equation in R n , n ≥ 7 . It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity. [ABSTRACT FROM AUTHOR]

We consider the equation 1 Δ x u + u yy + f (u) = 0 , x = (x 1 , ⋯ , x N) ∈ R N , y ∈ R , where N ≥ 2 and f is a sufficiently smooth function satisfying f (0) = 0 , f ′ (0) < 0 , and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in x ′ = (x 1 , ... , x N - 1) and decaying as | x ′ | → ∞ , periodic in x N , and quasiperiodic in y. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator with gradient terms. By using the direct method of moving planes, we prove the symmetry and monotonicity for positive solutions to the fractional p-Laplacian system in bounded domains and in the whole space. In the proof, decay at infinity and boundary estimate are key ingredients. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to analyze the existence and regularity of solutions for the following nonlocal parabolic problem involving the fractional Laplacian with singular nonlinearity u t + (- Δ) s u = f (x , t) u γ in Ω T : = Ω × (0 , T) , u = 0 in (R N \ Ω) × (0 , T) , u (· , 0) = u 0 (·) in Ω , where Ω is a bounded domain of class C 0 , 1 in R N , N > 2 s with s ∈ (0 , 1) , γ > 0 , 0 < T < + ∞ , f ≥ 0 , f ∈ L m (Ω T) , m ≥ 1 , is a non-negative function on Ω T and u 0 is a non-negative function defined on Ω belonging to some Lebesgue spaces. The existence and regularity of the very weak and weak solutions are obtained under different assumptions on the summability of the data and on γ . One of the difficulties arising in the problem is the proof of the strict positivity of the weak solutions inside the parabolic cylinders. This proof involves the smallest eigenvalue of the fractional Laplacian and the weak comparison principle. We also deal with the case of singular Radon measures data and the asymptotic behavior of the very weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff-type equation: - a + b ∫ R N | ∇ u | 2 d x Δ u + λ V (x) u + μ ϕ | u | p - 2 u = | u | q - 2 u , in R N , (- Δ) α 2 ϕ = | u | p , in R N , where a , b > 0 , 0 < α < N , N ≥ 3 , 2 ≤ p < N + α N - 2 , 2 < q < min { 4 , 2 ∗ } , 2 ∗ = 2 N / (N - 2) and λ , μ > 0 are parameters. Using the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for the above problem when b , μ are sufficiently small, and λ is large enough. Furthermore, the decay rate and the asymptotic behavior of the positive solutions are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Caffarelli–Kohn–Nirenberg (CKN) inequality: (∫ R N | x | - b (p + 1) | u | p + 1 d x) 2 p + 1 ≤ C a , b , N ∫ R N | x | - 2 a | ∇ u | 2 d x where N ≥ 3 , a < N - 2 2 , a ≤ b ≤ a + 1 and p = N + 2 (1 + a - b) N - 2 (1 + a - b) . It is well-known that up to dilations τ N - 2 2 - a u (τ x) and scalar multiplications Cu(x), the CKN inequality has a unique extremal function W(x) that is positive and radially symmetric in the parameter region b FS (a) ≤ b < a + 1 with a < 0 and a ≤ b < a + 1 with a ≥ 0 and a + b > 0 , where b FS (a) is the Felli–Schneider curve. We prove that in the above parameter region the following stabilities hold: stability of CKN inequality in the functional inequality setting d i s t D a 1 , 2 2 (u , Z) ≲ ‖ u ‖ D a 1 , 2 (R N) 2 - C a , b , N - 1 ‖ u ‖ L p + 1 (| x | - b (p + 1) , R N) 2 where Z = { c W τ ∣ c ∈ R \ { 0 } , τ > 0 } ; stability of CKN inequality in the critical point setting (in the class of nonnegative functions) d i s t D a 1 , 2 (u , Z 0 ν) ≲ Γ (u) , p > 2 or ν = 1 , Γ (u) | log Γ (u) | 1 2 , p = 2 and ν ≥ 2 , Γ (u) p 2 , 1 < p < 2 and ν ≥ 2 , where Γ (u) = ‖ d i v (| x | - a ∇ u) + | x | - b (p + 1) | u | p - 1 u ‖ H - 1 and Z 0 ν = { (W τ 1 , W τ 2 , ... , W τ ν) ∣ τ i > 0 }. Our results generalize the recent work in [7, 11, 15] on the sharp stability of profile decompositions for the special case a = b = 0 (the Sobolev inequality) to the above parameter region of the Caffarelli–Kohn–Nirenberg inequality. This parameter region is optimal for such stabilities in the sense that in the region a < b < b FS (a) with a < 0 , the nonnegative solution of the Euler–Lagrange equation of CKN inequality is no longer unique. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the existence or non-existence of solutions for a class of elliptic problems involving the N-Laplacian operator in the whole space. The nonlinearity considered involves critical Trudinger–Moser growth. Our approach is non-variational, and in this way we can address a wide range of problems not yet contained in the literature. Even W 1 , N (R N) ↪ L ∞ (R N) failing, we establish ‖ u ‖ L ∞ (R N) ≤ C ‖ u ‖ W 1 , N (R N) Θ (for some Θ > 0 ), when u is a solution. To conclude, we explore some asymptotic properties. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following autonomous Choquard equation − Δ u + u = (I α ∗ F (u)) F ′ (u) in R N , where N ≥ 3 , I α denotes the Riesz potential of order α ∈ (0 , N) and F satisfies general critical growth conditions. By using the variational methods and the Pohožaev manifold techniques, we prove the existence of radially symmetric positive ground state solution. [ABSTRACT FROM AUTHOR]

This paper deals with the following fractional Schrödinger equation with critical growth (- Δ) s u + V (x) u = | u | 2 s ∗ - 2 u , x ∈ Ω , u = 0 , x ∈ R N \ Ω , where Ω = R N or Ω is an open bounded domain of R N , N > 2 s with s ∈ (0 , 1) and V(x) is a sign-changing function. Firstly, using a nonlocal version of the second concentration-compactness principle, we prove the existence of Mountain-Pass solution for the above equation in R N . Secondly, we prove the existence of N distinct pairs of nontrivial solutions for the above equation in R N by using a global compactness result and Krasnoselskii’s genus theory, as well as in bounded domains. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non‐linearities on an unbounded domain, and show that such an equation with positive or sign‐changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

We investigate a fractional Schrödinger-Poisson system with strong singularity as follows: (−Δ)su+V(x)u+λϕu=f(x)u−γ,x∈R3,(−Δ)tϕ=u2,x∈R3,u>0,x∈R3,\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+V\left(x)u+\lambda \phi u=f\left(x){u}^{-\gamma },& x\in {{\mathbb{R}}}^{3},\\ {\left(-\Delta )}^{t}\phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\\ u\gt 0,& x\in {{\mathbb{R}}}^{3},\end{array}\right. where 034s+2t\gt 3, λ>0\lambda \gt 0 and γ>1\gamma \gt 1. When VV and ff satisfy certain conditions, existence and uniqueness of positive solution uλ{u}_{\lambda } are established via variational method and Nehari method. We also describe the asymptotic behaviour of uλ{u}_{\lambda } as λ→0\lambda \to 0.

We study the following Kirchhoff type equation:

We investigate the ground states for the focusing, subcritical nonlinear Schrödinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. A refinement of a classical result on rearrangements is proved to obtain qualitative features of the ground states. [ABSTRACT FROM AUTHOR]

We are concerned with the half-space Dirichlet problem - Δ v + v = | v | p - 1 v in R + N , v = c on ∂ R + N , lim x N → ∞ v (x ′ , x N) = 0 uniformly in x ′ ∈ R N - 1 , where R + N = { x ∈ R N : x N > 0 } for some N ≥ 2 , and p > 1 , c > 0 are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number c p ∈ (1 , e) , depending only on p, such that for 0 < c < c p there are infinitely many bounded positive solutions, whereas, for c > c p there are no bounded positive solutions. They also posed as an interesting open question whether the one-dimensional solution is the unique bounded positive solution in the case where c = c p . If N = 2 , 3 , we recently showed this one-dimensional symmetry property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the proof of De Giorgi’s conjecture in low dimensions. Here, we first focus on the case 1 < p < 3 and prove this uniqueness property in dimensions 2 ≤ N ≤ 5 . Then, for the cubic NLS, where p = 3 , we establish this for 2 ≤ N ≤ 4 . Our approach is completely different and relies on showing that a suitable auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in R N - 1 , for which an optimal Liouville type result is available. [ABSTRACT FROM AUTHOR]

We consider the supercritical Hénon problem - Δ u = C (α) | x | α u p α - ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where p α = N + 2 + 2 α N - 2 , α > 0 , Ω ⊂ R N is a smooth bounded domain containing the origin, C (α) = (N + α) (N - 2) and N ≥ 3 . The existence single peaked solutions has been established by Gladiali and Grossi in [J. Differential Equation 253, 2616–2645 (2012)]. In this paper, we investigate qualitative property of the single peaked solutions for small | ε | . More precisely, we obtain the non-degeneracy and uniqueness of the single peaked solutions for ε > 0 small enough. Moreover, if ε < 0 small, we show that above problem has no single peaked solutions. It is known that when α = 0 , the non-degeneracy and uniqueness of the single peaked solutions will depend on the Robin function. However, if α > 0 , the role of Robin function will disappear because of the presence of the weighted term | x | α . Also our results indicate that the Hénon critical exponent p α = N + 2 + 2 α N - 2 plays an essential role in the Hénon equation as the critical exponent N + 2 N - 2 in the case α = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear Choquard equation - Δ u + u = | x - y | - μ ∗ G (u) g (u) , in R N , where 0 < μ < min { N , 4 } , N ≥ 3 , g (u) ∈ C (R , R) satisfies very general critical growth conditions in the sense of the Hardy–Littlewood–Sobolev inequality and G (u) = ∫ 0 u g (s) d s . Using the Pohozaev constraint, we find that the above problem admits a ground state solution of Pohozaev type. [ABSTRACT FROM AUTHOR]

We apply the finite-dimentional reduction method to study a critical Grushin problem with competing potentials and we construct infinitely many positive solutions with multi-bubbles for it. Particularly, we resort to the local Pohozaev identities to locate the concentration points of these solutions instead of estimating the derivatives of the reduced functional, since the concentration points can be saddle points. Moreover, we discover that the stable critical point of the potential in nonlinear term plays a more important role than that of the potential in linear term in the process of constructing solutions. [ABSTRACT FROM AUTHOR]

We consider the following problem with combined nonlinearities: { − Δ u + a (y) u + b (y) u q − u p = 0 , u > 0 , in R N , u ∈ H 1 (R N) , where 1 < p < 2 ∗ − 1 , q > 1 , 2 ∗ = 2 N N − 2 when N ≥ 3 ; 2 ∗ = + ∞ when N = 2. By use of the Lyapunov-Schmidt reduction argument, under some expansion conditions of the potentials a (y) and b (y) , we establish infinitely many non-radial solutions. Especially, the nonlinearity b (y) u q is allowed to be (super-)critical, that is, q ≥ 2 ∗ − 1. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the 'charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when 'charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study existence and regularity results for the solution to a nonlinear singular parabolic problems involving Hardy potential ∂ u ∂ t - div a (x , t , ∇ u) - μ u p - 1 | x | p = f (x , t) u γ in Ω × (0 , T) , u = 0 on ∂ Ω × (0 , T) , u (x , 0) = u 0 (x) in Ω , where Ω is a bounded open subset on R N , N ≥ 3 , 0 ∈ Ω and γ > 0 , 2 ≤ p < N , 0 < T < + ∞ , μ > 0 , 0 ≤ f ∈ L m (Q) , m ≥ 1 and u 0 ∈ L ∞ (Ω) satisfies ∀ w ⊂ ⊂ Ω , ∃ M w > 0 : u 0 ≥ M w in ω. [ABSTRACT FROM AUTHOR]

We study Choquard type equation of the form where N ≥ 3 , I α is the Riesz potential with α ∈ (0 , N) , p > 1 , q > 2 and ε ≥ 0 . Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of (P 0) and of (P ε) with ε > 0 . We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to (P ε) . In the second part of the paper, for ε → 0 we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of (P ε) in each of the regimes. We also outline three different asymptotic regimes in the case ε → ∞ . In one of the asymptotic regimes positive groundstates of (P ε) converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of (P ε) with α = 0 . In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate. [ABSTRACT FROM AUTHOR]

The Generalized Relative Entropy inequality is a ubiquitous property in mathematical models applied in physics or biology. In spite of its importance, it is currently proved on a case-by-case basis in the literature. Here, we show that GRE is actually a generic consequence of Łoskot-Rudnicki’s inequality that is reminiscent of Jensen’s inequality. [ABSTRACT FROM AUTHOR]

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation: { u + V (x) u = u log u 2 i n R N , u H 1 (R N) , where N ⩾ 1, ⋋ > 0 is a parameter and the nonnegative continuous function V: ℝN → ℝ has potential well Ω:= int V−1(0) which possesses k disjoint bounded components Ω= ∪ j = 1 k Ω j . Using the variational methods, we prove that if the parameter ⋋ > 0 is large enough, then the equation has at least 2k − 1 multi-bump positive solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system. [ABSTRACT FROM AUTHOR]

In this paper, we establish concentration and multiplicity properties of ground state solutions to the following perturbed double phase problem with competing potentials: - ϵ p Δ p u - ϵ q Δ q u + V (x) (| u | p - 2 u + | u | q - 2 u) = K (x) f (u) , in R N , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , u > 0 , in R N , where 1 < p < q < N , Δ s u = div (| ∇ u | s - 2 ∇ u) , with s ∈ { p , q } , is the s-Laplacian operator, and ϵ is a small positive parameter. We assume that the potentials V, K and the nonlinearity f are continuous but are not necessarily of class C 1 . Under some natural hypotheses, using topological and variational tools from Nehari manifold analysis and Ljusternik–Schnirelmann category theory, we study the existence of positive ground state solutions and the relation between the number of positive solutions and the topology of the set where V attains its global minimum and K attains its global maximum. Moreover, we determine two concrete sets related to the potentials V and K as the concentration positions and we describe the concentration of ground state solutions as ϵ → 0 . The asymptotic convergence and the exponential decay of positive solutions are also explored. Finally, we establish a sufficient condition for the non-existence of ground state solutions. [ABSTRACT FROM AUTHOR]

In this article, we consider positivity issues for the clamped plate equation with high tension γ > 0 . This equation is given by Δ 2 u - γ Δ u = f under clamped boundary conditions. Here, we show that given a positive f, i.e. upwards pushing, we find a γ 0 > 0 such that for all γ ≥ γ 0 the bending u is indeed positive. This γ 0 only depends on the domain and the ratio of the L 1 and L ∞ norm of f. In contrast to a recent result by Cassani and Tarsia, our approach is valid in all dimensions. [ABSTRACT FROM AUTHOR]

We consider ground states of the nonlinear fractional Schrödinger equation with potentials

In this study, we consider the following quasilinear Choquard equation with singularity −Δu+V(x)u−uΔu2+λ(Iα∗∣u∣p)∣u∣p−2u=K(x)u−γ,x∈RN,u>0,x∈RN,\left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where Iα{I}_{\alpha } is a Riesz potential, 00\lambda \gt 0. Under suitable assumption on VV and KK, we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ→0\lambda \to 0.

We systematically inquire positive radial solutions of the quasilinear elliptic equation - Δ p ϕ = | x | σ ϕ q with Δ p ϕ : = div (| ∇ ϕ | p - 2 ∇ ϕ) is the p-Laplace operator, q > p - 1 , σ > - p and 1 < p < N . It is known that the radial solutions of this equation can be classified into three different types: the M-solutions (singular at r = 0 ), the E-solutions (regular at r = 0 ) and the F-solutions (whose existence begins away from r = 0 ). For these radial solutions, we find a substitution which can transfer this equation into an autonomous Lotka–Volterra system, such that the F-, E- and M-solutions can be comprehensively characterized. In particular, the M-solution has extremely plentiful properties, and its asymptotic expansions are more complicated. For the subcritical case of q, by virtue of a priori estimates, we employ an iterative method which can improve the accuracy step by step, to derive their precise asymptotic expansions. For the supercritical case of q, with the help of the autonomous Lotka–Volterra system and an inverse transformation, we obtain the asymptotic expansions of the M-solutions near the origin. In the same manner, we could acquire more accurate asymptotic expansions of the E-solutions via the iterative method whereas the F-solutions can be described by the autonomous Lotka–Volterra system. [ABSTRACT FROM AUTHOR]

In this paper, we consider fully nonlinear parabolic problems involving the fractional Laplacian. Hopf type lemmas for both on bounded domain Ω with smooth boundary and half space are obtained. When Ω is a ball or the whole space, we obtain the radial symmetry results of positive solutions. Our results are an extension of Li-Nirenberg [19], Li-Chen [18] and Wang-Chen [24]. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple normalized solutions to the following class of elliptic problems - Δ u = λ u + h (ϵ x) f (u) , in R N , ∫ R N | u | 2 d x = a 2 , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, h : R N → [ 0 , ∞) is a continuous function, and f is continuous function with L 2 -subcritical growth. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. [ABSTRACT FROM AUTHOR]